WWW 2011 – Session: Information Spread
March 28–April 1, 2011, Hyderabad, India
Milgram-Routing in Social Networks
Silvio Lattanzi
∗
Google, Inc.
76 Ninth Avenue
New York, NY
[email protected]
Alessandro Panconesi
Sapienza University of Rome
113 Via Salaria
Rome, Italy
[email protected]
ABSTRACT
D. Sivakumar
‡
Yahoo!
4401 Great America Parkway
Santa Clara, CA 95054, USA
[email protected]
the letter along to a friend, with the same instructions. The
surprising outcome was that a reasonably large fraction of
the letters reached the target and moreover, they did so in
very few hops. The success of Milgram’s experiments led to
the fascinating small world hypothesis: take any two people
in a social network, and they will be connected by a short
chain of acquaintances. The extent to which the hypothesis
is true is still actively debated. In this paper we give new
experimental and theoretical results concerning Milgram’s
experiment.
Empirical results: We perform an “in silico” replica of
the experiment where a cognitive “space of interests” is navigated. In our experiment we consider a social network of
co-authorships of computer science papers. Two people in
this networks are “friends” if they are co-authors. We then
extract a space of interests consisting of computer science
topics. In simulating the experiment, we go from person to
person by moving to the friend of the current person that
has more interests in common with the target. To the best
of our knowledge, this is the first time that the concept of
an interest space is used in a pure digital replica. Previous
studies such as [25, 30] made use of geographical proximity
only (the move is toward the friend that is geographically
closest to the target), an overly constrained rendition of the
experiment.
Theoretical results: Motivated by the experiment above,
we give a new evolutionary model that captures the small
world phenomenon together with other important properties of real-world social networks. Among these there are
dynamic properties, such as densification and shrinking diameter, and static properties like heavy-tailed distribution
of popularity (number of friends). The main aspect of the
model is its built-in “interest space,” a space of concepts that
is navigable.
An important issue with Milgram’s small world hypothesis is the difficulty of its verification. Milgram’s painstaking
work enabled him to collect data on a few hundreds of individuals. The advent of the internet has made it possible
to perform large-scale replicas of the experiment, as in [6],
or genuine “in silico” experiments, where there is no human
participation— the experiment is made only with social data
in digital form. Such “genuine” digital replicas are the main
focus of this paper. One such instance is the study in [30].
A snapshot of the social networking site LiveJournal was
downloaded to obtain a social network of roughly 15 million individuals. The experiment was simulated by picking
source and target at random, and by moving toward the target according to geographical proximity (geo-greedy): from
We demonstrate how a recent model of social networks (“Affiliation Networks”, [21]) offers powerful cues in local routing
within social networks, a theme made famous by sociologist Milgram’s “six degrees of separation” experiments. This
model posits the existence of an “interest space” that underlies a social network; we prove that in networks produced by
this model, not only do short paths exist among all pairs of
nodes but natural local routing algorithms can discover them
effectively. Specifically, we show that local routing can discover paths of length O(log2 n) to targets chosen uniformly
at random, and paths of length O(1) to targets chosen with
probability proportional to their degrees. Experiments on
the co-authorship graph derived from DBLP data confirm
our theoretical results, and shed light into the power of one
step of lookahead in routing algorithms for social networks.
Categories and Subject Descriptors
G.3 [Mathematics of Computing]: Probability and statistics—Stochastic processes
General Terms
Theory
Keywords
Social Networks, Affiliation Networks, Milgram’s experiment
1.
†
INTRODUCTION
Milgram’s six-degrees-of-separation experiment [28] and
the fascinating small world hypothesis that follows from it
have been rich sources of interesting research in recent years.
In this landmark experiment, human subjects were asked to
deliver a letter to a target person in a far-away city, following
a simple rule: If they knew the target on a first name basis,
they would deliver the letter; otherwise, they would pass
∗Research done during an internship at Google when the
author was in the Dipartimento di Informatica of Sapienza
University
†The author is supported by a Google Research Award and
a Yahoo! Faculty Award
‡Research conducted while author was at Google Inc.
Copyright is held by the International World Wide Web Conference Committee (IW3C2). Distribution of these papers is limited to classroom use,
and personal use by others.
WWW 2011, March 28–April 1, 2011, Hyderabad, India.
ACM 978-1-4503-0632-4/11/03.
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the current node X we move to the neighbor of X that is
closest to the target. In another instance [25], the diameter
of the social network of IM chat exchanges was estimated
and found to be compatible with the small world hypothesis. These “cyber-replicas” have the obvious advantage that
one can test the small-world hypothesis with millions of individuals. The main problem is how to make them realistic.
The limitation of the approaches in [25, 30] is that they
only take into account geographical or positional information, while it is clear that cognitive cues play a role in the
original experiment. As mentioned, in this paper we present
a cyber-replica of Milgram’s experiment in which a “space of
concepts” is navigated. In simulating the experiment, we go
from person to person by moving to the friend of the current
person that has more interests in common with the target.
Our experiments strongly reinforce two significant pieces of
work in the sociology literature — the importance of weak
ties [12] and the significance of the social status of the target node in Milgram’s experiment [17]. Finally, since our
experiments are based on publicly available data, it should
be possible for other researchers to replicate our work as well
as derive additional insights underlying small-world routing.
There is a rich literature on stochastic models that reproduce several salient features of real-world social networks
(e.g. power law distributions of popularity [5, 8], high clustering coefficient [26], densification and shrinking diameter [24]). There is, however, no model that seamlessly captures all of them. It would also be nice to have an evolutionary, as opposed to static, model of small world, where
the network is evolving as new entrants join instead of being fixed. And of course, most interestingly, such a model
should have a natural notion of “space of interests” that is
navigable and that co-evolves with the social network. In
this paper we do precisely this, by introducing a new evolutionary model that builds on the work in [21] and that
captures nicely many salient sociological properties of realworld networks.
The model is based on affiliation networks, a concept first
introduced in sociology by Breiger [4] and extended in [21].
An affiliation network is a bipartite graph with people on
one side and interests on the other. The affiliation network
comes with an associated friendship graph in which two people are friends if they share an interest. In the friendship
graph people can also become friends because of “popularity” of one of the two parties, that is, preferential attachment
is also in effect. In the original model people and interests
join dynamically [21]. Each new node (person or interest)
is a random perturbation of one pre-existing node. In our
new model, a new interest that joins the network can be a
perturbation of a mixture of pre-existing interests and, similarly, a new person joining the network will share a subset
of the interests of several friends, as opposed to just one of
them. Thus, this extension is more natural. While this is
a small variation of the original model, the new model exhibits several interesting new properties (that are not known
to be enjoyed by the original model). Our model is the first
to exhibit simultaneously three different (sets of) properties of social networks: small world phenomena, evolutionary properties, and navigability of the interest space. In
previous attempts, these features were somehow captured
but separately. For instance, the models in [9, 10, 16, 35]
deal with the small world phenomenon, but they are static
and unable to explain evolutionary properties or even the
heavy-tailed distribution of popularity (number of friends).
Furthermore they assume that every person knows the distance between its neighbors and the target, while we only
assume that every person knows how similar interests are.
There have been also some attempts to define and navigate
an interest space instead of geographic information [15, 34]
or to use a latent space of interests to define the friendship
graph [31, 33]. But, again, these models are static (the number of nodes in the graph does not increase with time) and
unable to explain evolutionary properties. In contrast, in
our proposal all these different aspects come forth naturally
from the same simple model.
Our enhanced model has several strong properties that
are especially relevant for modeling small worlds, matching
the experimental evidence from a quantitative point of view.
The effective diameter of the friendship graph is bounded
from above by a constant. This is compatible with the empirical observations of [25] where a very large social-network
of hundreds of millions of nodes was analyzed, and its effective diameter found to be a very small number. When we
analyze the actual working of Milgram routing in the friendship graph (not to be confused with the mere existence of
short paths), we find that when source and target are chosen at random, their expected routing distance is O(log2 n).
The novelty here is that to find this short chain we navigate
the interest space associated with the affiliation network,
and not the friendship graph itself. When the target is chosen by popularity, i.e. with probability proportional to the
numbers of friends, then the expected length of the chain
can be upper bounded by a constant. This is in line with
the experimental evidence with human subjects. It has been
pointed out that the successful outcome of Milgram’s experiment could depend on the fact that the target was a person
of high social status and had a profession that contributed
even more than his status to establish and nurture many
social connections [17]. Our model captures these features
of the real world very nicely. Further, in accordance with
the observation of Granovetter [12], the proofs of the upper
bound for the diameter and the expected routing distance
use heavily the presence of weak ties (i.e. preferential attachment edges in the model). To summarize, our analysis
shows that our model incorporates not only basic structural
facts of real-world networks, but can also explain some of
their more nuanced features.
1.1
Related Work
We now overview the most relevant literature. Local routing algorithms have been intensely studied in the context of
distributed systems. In this context there are some attempts
to use the intuition behind the Milgram’s experiments to
build new algorithms. Besides the work in [25, 30] that
make use of social networks, other authors have replicated
Milgram’s experiment in the real world [13] or by using using email [2, 6]. These experimental findings are compatible
with the small world hypothesis. The issue of attrition, the
natural tendency of human subjects to drop out of the experiment, is analyzed in [11]. This social attrition introduces a
bias in favor of short chains, because long chains tend to be
interrupted before reaching the target. Taking this bias into
account makes chains somewhat longer on average. Other
interesting critiques to the Milgram’s experiment are presented in [17].
From a theoretical viewpoint, one of the first observations
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that led to the interest in random graph models significantly
different from the classical Erdős–Rényi models comes in the
work of Faloutsos et al.[8], who noticed that the degree distribution of the Internet graph (the graph whose vertices
are computers and whose edges are network links) is heavytailed, and roughly obeys a “power law,” that is, for some
constant α > 0, the fraction of nodes of degree d is proportional to d−α . Similar observations were made about the
web graph (the graph whose vertices are web pages, and
whose directed edges are hyperlinks among web pages) by
Barabasi and Albert [3], who also presented models based
on the notion of “preferential attachment,” wherein a network evolves by new nodes attaching themselves to existing
nodes with probability proportional to the degrees of those
nodes. Both works draw their inspiration and mathematical
precedents from classical works of Zipf [36], Mandelbrot [27],
and Simon [32]. Later Broder et al. [5] made a rich set of
observations about the degree and connectivity structure of
the web graph, and showed that besides power-law degree
distribution, the web graph consisted of numerous dense bipartite subgraphs (often dubbed “communities”). Aiello et
al. [1] and Kumar et al. [19] presented three models of random graphs that offer rigorous explanations for power-law
degree distributions.
After the discovery of some surprising evolutionary properties such as densification and shrinking diameter in [24],
several new models have been introduced [24, 22, 23] but
none of them could really explain the new properties before
the introduction of the affiliation network model [21]. The
affiliation network model is the first model where interests
have a crucial role and so is the first evolution model where it
is possible to study Milgram’s experiment. In two previous
papers [20, 29], the connectivity and the degree distribution
of a static version of affiliation network model have been
studied.
The works of Watts and Strogatz [35] and of Kleinberg [16]
are the closest in spirit to ours in that they offer graph models that incorporate natural routing algorithms. In Kleinberg’s model, vertices reside in some metric space, and a
vertex is usually connected to most other vertices in its metric neighborhood, and, in addition, to a few “long range”
neighbors. He proved the remarkable result that the network has small diameter and easily discoverable paths iff the
long-range neighbors are chosen in a specific way. Kleinberg’s models offer a nice starting point to analyze social
networks, but because of its stylized nature, isn’t applicable in developing an understanding of the structure of real
social networks. The other limitation of Kleinberg’s model
is that it is static, and is not a model of graph evolution.
For this reason several extensions of Kleinberg’s model have
been introduced in order to study the problem starting from
a different initial topology [10] or adding some constraint on
the final degree distribution of the graphs [9].
2.
P1
I1
P2
I2
P3
I3
P2
P1
P3
P1
I1
P2
I2
P3
I3
P2
P1
P3
P4
P4
(A)
P1
I1
P2
I2
P3
I3
P2
P1
P3
(B)
P1
I1
P2
I2
P3
I3
P2
P1
P3
P4
P4
P4
P4
(C)
P1
I1
P2
I2
P3
I3
P2
P1
P3
(D)
P1
I1
P2
I2
P3
I3
P2
P1
P3
P4
P4
(E)
P4
P4
(F)
Figure 1: Insertion of a new person in the affiliation network and the social network derived from
it. (A)The initial affiliation network and the related
social graph. (B)Insertion of P4 in the affiliation network. (C)P4 selects as prototype P3 . (D)P4 copies a
perturbation of the edges of P3 . (E)The social graph
is updated. (F)P4 adds some preferential attachment
edges in the social graph.
people. In this graph, people can be friends for two different
reasons: if they share an interest or because of preferential
attachment. Thus, G is the “folding” of B, plus a set of edges
generated by preferential attachment. In [21] the graph B
evolves as follows. When a new interest (resp. person) comes
in, it selects a prototype node among the existing interests
(resp. people) and copies it with a small perturbation. In
this new version, when a new node joins B it can select more
than one prototype. A new interest for example, will be a
slightly perturbed mixture of a few existing interests, and a
new person will be interested in a combination of interests
of his/her friends. This new model seems more realistic and,
from the technical point of view, it presents a few complications that make it a non straightforward extension of the
previous one. More importantly, in this new version of the
model it is possible to prove that it enjoys some interesting additional properties. Figure 1 describes the insertion
of a new person in the affiliation network and the friendship
network. Table 1 describes the model precisely. For readability, we present the two evolution processes separately
even though the two graphs evolve together.
3.
PRELIMINARIES
We say that an event occurs with high probability (whp)
if it happens with probability 1 − o(1), where the o(1) term
goes to zero as n, the number of vertices, goes to ∞. A
random variable X is said to be heavy-tailed if
limx→∞ eλx P r[X > x] = ∞ for all constants λ > 0.
OUR MODEL
The model that we consider in this paper is a variation of
the one presented in [21]. In both models, two graphs evolve
at the same time. The first is a bipartite graph, denoted as
B(P, I), that represents the affiliation network, with a set P
of people on one side and a set of interests I on the other. An
edge (p, i) represents the fact that p is interested in i. The
second graph is a friendship network, denoted as G(P, E),
representing friendship relations within the same set P of
Definition 1. The graph G(P, E) will be referred to as
the friendship graph. An edge of G between two people that
comes from the fact that they share an interest in B is called
a folded edge.
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B(P, I)
Fix
Pk2 k1 and k2 , fix k + s integers
Pk1 two integers
c
=
c
,
p
p
j
j=1 cij = ci > 0, and let β ∈ (0, 1).
j=1
At time 0, the bipartite graph B0 (P, I) is a simple
graph with at least cp ci edges, where each node in P
has at least cp edges and each node in I has at least
ci edges.
At time t > 0:
(Evolution of P ) With probability β:
(Arrival ) A new node p is added to P .
(Preferentially chosen Prototypes) A set of nodes
p1 , · · · , pk1 ∈ P , with k > 1, are chosen as prototypes for the new node, with probability proportional
to their degrees.
(Edge copying) cpj edges are “copied” from pj , with
Pk1
1 ≤ j ≤ k1 and
j=1 cpj = cp ; that is, cpj neighbors of pj , denoted by i1 , . . . , icpj , are chosen uniformly at random (without replacement), and the
edges (p, i1 ), · · · , (p, icpj ) are added to the graph.
(Evolution of I) With probability 1 − β, a new node
i is added to I following a symmetrical process, adding
ci edges to i.
G(P, E)
Fix
Pk2 k1 , k2 and s, fix k + s integers
Pk1 three integers
c
=
c
,
p
p
j
j=1 cij = ci > 0, and let β ∈ (0, 1).
j=1
At time 0, G0 (P, E) consists of the subset P of the
vertices of B0 (P, I), and two vertices have an edge between them for every neighbor in I that they have in
common in B0 (P, I).
At time t > 0:
(Evolution of P ) With probability β:
(Arrival ) A new node p is added to P .
(Edges via Prototype) An edge between p and another
node in P is added for every neighbor that they have
in common in B(P, I) (note that this is done after the
edges for p are determined in B).
(Edges via evolution of I)
With probability 1 − β:
A new edge is added between two nodes p1 and p2 if
the new node added to i ∈ I is a neighbor of both p1
and p2 in B(P, I).
(Preferentially Chosen Edges) A set of s nodes
pi1 , . . . , pis is chosen, each node independently of the
others (with replacement), by choosing vertices with
probability proportional to their degrees, and the
edges (p, pi1 ), . . . , (p, pis ) are added to G(P, E).
Table 1: Description of the evolving model.
4.
BASIC PROPERTIES OF THE MODEL
where


In this section we show that the properties of the original
model in [21] are also enjoyed by the new model. We begin
by defining the concept of effective diameter that intuitively
measures the largest distance between “almost all” pair of
nodes in a graph.
1
γ<
4+
cp β
ci (1−β)
γ <
1
4+
ci (1−β)
cp β

.
(2) The degree distributions of the graphs G(P, E) is heavytailed with high probability.
(3) The number of edges in G(P, E) is ω(n) with high probability.
(4) The q-effective diameter of G(P, E) shrinks or stabilizes after time φn with high probability, for any constant
0 < φ < 1 and for any constant 0 < q < 1.
Definition 2. [Effective Diameter] For 0 < q < 1,
the q-effective diameter is the minimum de such that, for at
least a q fraction of the node pairs, the length of the shortest
path between the pair is at most de .
Then we define the core and hubs. Intuitively, they define
the popular interests and the people that are connected to
them.
Now we state two technical lemmas that we will use in the
following sections. The proofs are omitted from this extended abstract.
Definition 3. [Core and hubs] Let d(v) be the degree
of v. A set of interests C ⊆ I is an α-core of an affiliation
network B(P, I) if d(v) ≥ αn for all v ∈ C. The hubs are
the people in P at distance one from C.
Lemma 2. Let 1 > > 0 be any constant and let v be
a node in B(P, I) with degree g(n) at time n, with g(n) ∈
Ω(log2 n). Then, with high probability, v’s degree at time n
is smaller than C · g(n), for some constant C > 0. Furthermore, if a node v has degree o(log2 n) at time n or it
is inserted after time n, then the final degree of v is in
o(log2 n) with high probability.
In what follows we will refer to α-cores simply as cores.
We now list a set of properties that our model shares with
the original affiliation network of [21]. The proofs are similar
and omitted from this extended abstract. The properties
are the heavy tailed distribution of degrees in both B(P, I)
and G(P, E), and densification and shrinking diameter of
G(P, E).
Lemma 3. With high probability, any node of P inserted
after time φn, for any constant φ > 0, will be connected to
a hub via a preferential attachment edge. Further we have
that, with high probability for t > φn:
!
c (1−β)
X
1+γ 1− i c β
p
V (hubs, t) =
dG (v) ∈ Ω t
Theorem 1. [General properties of the model]
(1)Given an affiliation network B(P, I), the degree sequence
of nodes in P (resp. I), almost surely when n → ∞, follows
a power law distribution with exponent
v∈hubs
cp β
α = −2 −
ci (1 − β)
resp.α = −2 − ci (1−β)
, for every degree smaller than nγ
cp β
and
V (G/hubs, t) =
X
v ∈hubs
/
728
dG (v) ∈ Θ t
!
c (1−β)
1+ 1− i c β
p
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March 28–April 1, 2011, Hyderabad, India
Where γ and are two constants such that γ > and V (S, t)
is the volume of the node in S at time t.
5.
THE CRUCIAL ROLE OF WEAK TIES
In this section we study the effective diameter of G(P, E)
and show that it is bounded by a constant (it is unknown if
this property holds in the original affiliation network model).
This property is a consequence of the co-existence of folded
and preferential-attachment edges. Several studies have shown
that links in a social network are of two types, local and longrange, also called weak, ties [12]. Weak ties have several important structural properties, for instance they form bridges
between different communities and, in particular, they are
the crucial ingredient that makes small worlds possible.
In our model folded edges are local, for they connect people within a community of shared interests, while preferential attachment edges are the weak (or long-range) ties [12,
16]. Note that, in accordance with the previous literature
and sociological intuition, in our model weak ties are very
few compared to folded edges. In this section we show that
weak ties play another interesting structural function that is
in accordance with the empirical evidence: weak ties are crucial to bound the effective diameter of the friendship graph
by a constant.
Our proof also uses in a fundamental way the presence of
hubs. This might seem in contrast with the results in [6]
where the authors suggest that their role is not relevant. A
possible explanation is that they consider only the degree
induced by the explored paths, and thus consider only a
subgraph of the social network. Thus it is possible that in
their experiments a high degree node seems to have small
degree just because only few messages passed through him.
In our proof we consider the real degree of a node. We also
note that our results are in line with the original findings of
Milgram [27] and with our experiments.
Theorem 4. For every q < 1, there is a constant ∆q
such that the q-effective diameter of G(P, E) is bounded from
above above by ∆q .
The next lemma (proof omitted) on the distance between
nodes in the core of the affiliation network is crucial in the
proof of Theorem 4.
Lemma 5. Let C be the core of the affiliation network
B(P, I). There exists a constant d, independent of n, such
that, with high probability, the distance in the affiliation network between any pair of nodes in the core is at most d.
Corollary 6. Any two hubs are at constant distance in
G(P, E) and B(P, I), with high probability.
Proof. (of Theorem 4) Recall that from Lemma 3 we
have that all nodes in P inserted after time φn, for any
φ > 0, will have at least one preferential attachment edge
incident to a hub, with probability 1 − o(1). Now, let Xi be
a random variable such that:
1 if i has a hub in its neighborhood
Xi =
0 otherwise
Figure 2: An affiliation network(A) and the induced
social network(B) and hierarchy of interests(C). The
dotted lines from a to b in (A) represent that b is the
prototype of a.
φ. Observe that each Xi satisfies the Lipschitz condition
with di equal
concentration results [7] we
Pn1. So by standard
0
0
have that
i=φn Xi ≥ (1 − c )n, for any constant c > c.
Hence the claim follows from Corollary 6.
6.
LOCAL ROUTING IN INTEREST SPACE
In this section we analyze the performance of Milgram
routing in our model. It is clear that in Milgram’s experiment cues other than geographic distance play a role. For
instance, the target was defined not only by a location but,
crucially, by a profession. Therefore, if one wants more realistic models a more nuanced version of proximity must be
used. In this section we show that our model has a natural “space of interests” that is associated with the affiliation
network that is navigable. We note that it is not known if
the original affiliation network model enjoys the same property [21].
Two more aspects make the following analysis interesting
in our opinion. This is the first study of the performance
of local routing algorithm with an evolving model. Furthermore, ours is the first model that can explain Milgram’s experiment if we assume some constant attrition, as suggested
in [11] (i.e. in this case only paths of constant length can be
observed with high probability).
We start by defining a notion of distance between interests. In order to do so we define the prototype graph G(I, Ẽ).
The nodes of the prototype graph are the interests in the
affiliation network, and two interest i1 , i2 have an edge between them if i1 has been selected as a prototype for i2 or
vice versa. Furthermore, we have that two initial interests
i0 and i00 contained in the graph B0 (P, I) are connected if
there is a person in B0 (P, I) that is interested in (connected
to) both. Thus, the prototype graph consists of a clique of
the initial interests and of links connecting nodes to their
prototypes. In Figure 2 it is shown an example of affiliation network with the induced friendship network and the
prototype graph.
Definition 4. [Distance between interests] For two
nodes i1 , i2 ∈ I, we define the distance between i1 and i2
as the shortest (hop) distance between the two nodes in the
prototype graph. Further, we define the interest distance
between two people p1 and p2 as the smallest distance between
any interest of p1 and any interest of p2 .
The number of nodes
Pn that have
Pn at least one hub in their
neighborhood is
Xi ≥
i=1
i=φn Xi . From Lemma 3 it
hP
i
n
follows that E
X
≥
(1
− c)n, for any constant c >
i
i=φn
In our analysis we assume that every person is able to assess
the distance between any two interests. In practice we are
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March 28–April 1, 2011, Hyderabad, India
assuming that every person is able to compute the similarity
between any two interests, in order to decide which friend
is closest to the target. This natural assumption is made,
perhaps implicitly, in every previous navigation model. For
example in [16] a node is always able to select the neighbor
closest to the target in the metric space.
We define our routing algorithm as follows.
of a vertex is its degree, and that the volume of a set of
vertices is the sum of their volumes. Let hubs denote the set
of hubs. Let V (hubs, t) be the total volume of the hubs at
time t, and V (G/hubs, t) the total volume of the rest of the
graph at time t. As shown in Lemma 3 we have that, for
t > φn:
!
c (1−β)
X
1+γ 1− i c β
p
V (hubs, t) =
dG (v) ∈ Ω t
Definition 5. [Local Routing algorithm]
In each step the message holder u performs the following:
(1) If the destination is a neighbor of u, the message is
forwarded to it.
(2) Otherwise, u forwards the message to the neighbor that
minimizes the interest distance to the destination.
v∈hubs
and
V (G/hubs, t) =
X
dG (v) ∈ Θ t
!
c (1−β)
1+ 1− i c β
p
v ∈hubs
/
We start by proving a basic property of our algorithm.
Where γ > . Thus, when the destination is selected with
probability proportional to its degree, with probability 1 −
o(1), it will be a hub. In addition, Lemma 5 implies that two
hubs are within constant distance also in the interest space.
So, by Lemma 7, it holds with high probability that if a
message reaches a hub it will need only a constant additional
number of steps to reach every other hub using the local
routing algorithm.
Now note that Lemma 2 implies that all the hubs are
inserted before time φn with high probability, for every constant φ > 0. Further, by Lemma 3 every node inserted
after time φn will be connected to a hub with probability
1 − o(1). To summarize, with probability (1 − φ − o(1)), the
destination is a hub and the source has a hub in its neighborhood. It follows that the local routing algorithm will deliver
a message in a constant number of rounds, with probability
at least (1 − φ − o(1)).
Lemma 7. In every step of the local routing algorithm,
either the interest distance between the message holder and
the destination is reduced or the message is delivered to the
target.
Proof. If the message holder knows the target the lemma
is true by definition 5. Otherwise let v be any interest of the
message holder and let w(v) be an interest connected to v
in the prototype graph but with smaller distance from the
target. Note that w(v) always exists because the graph is
connected.
There are two cases: (i) if v, w(v) ∈ B0 (P, I) then there is
a person in B0 (P, I) interested to both v, w(v); (ii) if v is a
prototype of w(v) (or, symmetrically, vice-versa) then v and
w(v) have a neighbor in common in B(P, I) by definition
of the evolving process. In any case, for any interest v of
the message holder in the people graph, there is a person
interested in both v and w(v). It follows that in the neighborhood of the message holder, for any interest v, there is
a person interested in w(v). So using the local routing algorithm it is always possible to forward the message to the
neighbor closest to the target, and the claim follows.
We now consider a different setting. Suppose that we expand the interests of the destination in such a way that
they include the interest of its neighbors. We call this case
the expanded interests setting. This is an attempt to capture
the additional knowledge that human subjects have about
the destination, apart from its personal information. This is
interesting because it captures some features of the original
experiment. For instance, in the first experiment presented
by Milgram in [28], the sources knew also that the target
was married to a divinity student in Cambridge, MA.
In this setting we can prove the following. The proof is
similar to the proof of the previous Theorem and omitted
for lack of space.
We now show that for most source-destination pairs it is
possible to route the message within a constant number of
steps, provided that the destination is selected with a probability that is proportional to its degree, i.e. its “popularity”
in the social network. This result is in accordance with the
analysis of Milgram’s experiment done by Kleinfeld [17], who
pointed out that a successful outcome crucially depends on
the social status of the target1 .
β
cp . In the expanded interests
Theorem 9. Let ci < 1−β
setting when source and destination are selected uniformly
at random then, with probability (1 − 2φ − o(1)), the local
routing algorithm will route the message in constant many
steps, for every constant φ > 0.
β
Theorem 8. Let ci < 1−β
cp . If the destination is selected with probability proportional to its degree and the source
is selected uniformly at random then, with probability (1 −
φ−o(1)), for any constant φ > 0, the local routing algorithm
delivers the message in constant many steps.
Now we study the most general case, when source and target
are chosen adversarially and we do not extend the interest
space of the destination, in this setting we are able to show
the following upper bound on the running time of the local
routing algorithm.
Proof. Let v be the destination. We first prove that
with probability 1 − o(1) v is a hub. Recall that the volume
1
This point is in contrast with the claim in [6] but we recall
Kleinfeld’s observation in [17]. She wrote “Take the selection
of the sample. I found in the archives the original advertisement recruiting subjects for the Wichita, Kansas study. This
advertisement was worded so as to attract not representative
people but particularly sociable people proud of their social
skills and confident of their powers to reach someone across
class barriers.” Apart from this skepticism there are experiments suggesting that social barriers can actually hinder
Milgram’s local routing [18].
β
Theorem 10. If ci < 1−β
cp then, for any source and any
destination, the local routing algorithm routes the message
within O(log2 n) steps with high probability.
Proof. To prove the result we will bound the diameter
of the interest prototype graph. By Lemma 7 the diameter
is an easy upper bound for the delivery time of local routing.
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March 28–April 1, 2011, Hyderabad, India
We will show that, with high probability (whp), the diameter
of the prototype graph is O(log2 n).
The general idea of the proof is to divide the random
process in O(log n) macro-phases, and to show that in each
macro-phase
the probability that diameter increases by ω (log n)
is o
1
log n
we can upper bound τC as follows.
Pr[τC ]
. Thus, the diameter is O(log2 n) whp.
Let us divide the evolving process in O(log n) phases. In
phase zero we group the first 600 log n steps. Phase one is
from the end of phase zero to step b(1 + ) log nc, for a small
constant > 0. Phase two is up to step (1 + )2 log n . In
general,phase i starts after the end of phase i − 1 and ends
at step (1 + )i log n .
Let us now consider a generic phase t > 0. Let T =
(1 + )t 600 log n. First, we want a bound on the number of
edges in the affiliation network B(P, I) at the beginning of
each phase. Let At be the random variable that counts the
number of edges at the beginning of phase t. We have that
E[At ] = (βcp + (1 − β)ci )T . By the Chernoff bound,
≤
(# of steps in a phase) · (# of new nodes in a
phase) · P [xi (j) holds for node j of degree ci ]
e
≤ dT e dT
P [xi holds for a node j in a step]
C
cp c q C
dT e
2
≤ dT e C
T T −C
C
C
≤ dT e 1 +
(2cp cq )C
T
−
C
C
C
≤ dT eeC
(2cp cq )C
C C
< dT e (2cp cq ) ,
where in the third inequality we use Stirling’s approximation. Therefore the probability of τC decreases geometrically
with C.
Finally, let us compute the probability that the final diameter is greater than K = k log2 n. After the first phase the
diameter is at most 600 log n, so we can bound the previous
probability as the probability that the diameter increases by
at least (k − 600) log n after phase 1. Hence
X
log
n
Πi=2(1+) Pr[ξki ]
Pr[D ≥ k log n] ≤
E[At ]
1
1
E[At ] ≤ exp −
≤ 2.
P r |E[At ] − At | >
10
300
n
k2 ,k3 ,··· ,klog
(1+) n
Plog(1+) n
ki =K−600 log n
i=2
Using the union bound for the number of macro-phases, it
9
follows that at the beginning of each phase t, 10
E[At ] ≤
11
At ≤ 10 E[At ] with high probability. In the rest of the proof
9
we will assume that 10
E[At ] ≤ At ≤ 11
E[At ].
10
To get a bound on the diameter, we start by studying the
two following events:
≤
≤
ξ1 (j)
=
{interest j, inserted in phase t, of degree ci is
selected in a step during phase t as a prototype
for the first time}
ξ2 (j)
=
∈
{interest j, inserted in phase t, of degree ci
increases its degree in a step during phase t}
log(1+) n · (K − 600 log n) ·
· T log n · (2cp cq )K−600 log n
log(1+) n · (K − 600 log n) ·
·Θ nlog n · n−k log n
o(1)
Thus by choosing a large enough k the claim follows.
First notice that from the definition of the evolving process,
i
we have that Pr[ξ1 (j)] ≤ Acit ≤ 10c
.
9T
To bound Pr[ξ2 (j)], recall that interest j has degree ci ,
so there are ci people interested in it. Denote them as
p1 , p2 , · · · , pci . Now, if j increases its degree, it must be
because a new person joins the graph and copies the interest j from one of the person interested to it, p1 , p2 , · · · , pci .
This happens with probability:
Pr[ξ2 (j)] ≤
7.
EXPERIMENTS
Our mathematical model of social networks, building on
the affiliation network model, suggests natural decentralized routing algorithms in social networks. Namely, given
a source vertex s and a target vertex t, identify the interests of s and t in the underlying affiliation network and
identify the neighbor of s whose interests are closer to that
of t (with respect to the hierarchy of interests implied by
the prototype selection step). Inspired by this, one can define natural algorithms that perform decentralized routing
in real-world social networks by suitably approximating the
process of navigating the interest hierarchy. In this section,
we do precisely this, and report our findings based on simple
experiments with a modestly-sized social network.
Our social network consists of authors as nodes and edges
defined by co-authorship of one or more articles. We downloaded a copy of the DBLP database of computer science
papers, a DB of roughly 735,000 authors and 1.24M articles, and constructed the co-authorship graph with about
4.63M edges (for an average degree of roughly 6.7 co-authors
per node). On this network, we randomly selected about
575 pairs of source–target pairs and attempted to construct
paths between them. The largest connected component in
this network has roughly 80% of the vertices, with the rest
of the vertices in very small isolated components, so that
the probability that two randomly selected nodes belong to
c ci
X
10di
1 p
1− 1−
9T
di
i=1
Using calculus, it is possible to see that this probability is
maximized when d1 = · · · = dci = T . Thus
c cp cp ci
1 p
≤ ci 1 − e T ≤
Pr[ξ2 (j)] ≤ ci 1 − 1 −
T
T
c c
So Pr[ξ1 (j) ∨ ξ2 (j)] ≤ 2 pT i . Let us define ξ(j) := ξ1 (j) ∨
ξ2 (j).
Now we can compute the probability that in phase t the
diameter of the prototype graph increases by more then C,
with C > e. Let us call this event τC . Note that if τC holds,
a sequence of C new interests is added in phase t, increasing
the diameter of the prototype graph of C. In order for this
event to hold ξ has to occur at least C times in a phase. So
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March 28–April 1, 2011, Hyderabad, India
the largest connected component is roughly 64%. The mean
length of the shortest path between nodes in this component
is roughly 6.3 (with a median length of 6).
Notice that in this way, we construct an affiliation network
where two authors are friends if they coauthor a paper, now
we have to infer a metric on the interest in order to route
the messages. Unfortunately this is not easy, because there
is not a clear definition of closeness between papers and all
the standard classification system for the papers are too poor
for our purpose. To overcome this difficulty we define the
interest space not as the set of papers but as the set of
bigrams and unigrams contained in the title of the paper.
In particular we begin by segmenting article titles into
one-word and two-word sequences (unigrams and bigrams)
after suitably eliminating stopwords that occur commonly
(‘and’, ‘the’, etc.). For instance, the title “Small world experiments for everyone” generates four unigrams — ‘small’,
‘world’, ‘experiments’, and ‘everyone’, and two bigrams —
‘small world’, ‘world experiments’. Both bigrams and unigrams are treated as interests, with the latter of a more
generic kind; for instance, the unigram ‘physics’ is somewhat general, whereas the bigram ‘particle physics’ is much
more specific. In this fashion, for every author, their interest
profile is identified; specifically, for author a and interest i,
we define s(i, a) to be the strength of interest i for author
a, and is defined as the number of occurrences of interest
(unigram/bigram) i within author a’s publications.
To simulate Milgram’s experiment, our basic algorithm
operates as follows: if we are currently at node x, we move
to the neighbor y of x whose interest profile is closest to
the target t, where the measure of proximity of y to t is
computed according to the formula
proximity(y, t) =
Success Rate without expanded interests
100
Lookahead Monotone
Local Monotone
Lookahead
Local
Success Rate
80
60
40
20
0
0
2
4
6
8
10
12
Minimum degree of the destinations
14
16
Figure 3: Success Rate without extended interests.
s(i, y)s(i, t)
,
p(i)
X
Interest
ations of the four algorithms described above are LocalExpand, Local-Monotone-Expand, and so on.
Figure 3, 4 report the percentage of succesful chains for
the eight variations of the decentralized routing algorithm
we studied. For reference, we compare the performance of
the decentralized routing algorithms to that of the omniscient algorithm that has full information about the network
structure and employs a standard ‘shortest path’ computation. The ‘success percentage’ in Figure 3, 4 is the percentage of source–target pairs successfully routed, divided by
0.64 (which is the fraction for this omniscient algorithm).
The results are presented in four groups, each corresponding to one value of a parameter called τ , which restricts
the sampling of the target nodes to be uniform among all
nodes of degree at least τ ; this is done to explore the role
of the centrality of the target in determining the success of
decentralized routing.
i
Success Rate with expanded interests
where p(i) denotes
P the overall popularity of interest i, defined by p(i) = a s(i, a). If there is no neighbor with nonzero proximity, we either declare failure, or in a variation of
the experiment, proceed greedily to the neighbor of highest
degree.
The most basic variant of the algorithm outlined insists
that the proximity measure strictly increase in each step of
the routing: this version is called Local-Monotone, and
the version without this restriction is called Local. The
next variation we consider is to allow one step of ‘lookahead’,
where we not only evaluate neighbors of x, but also evaluate
neighbors of neighbors of x, and route through the neighobor whose neighbor achieves the highest proximity to the
target; this idea of ‘lookahead’, very common in computer
science, captures the belief that in real social networks, one
not only has knowledge about their friends, one often has
partial knowledge about friends-of-friends. The corresponding non-monotone and monotone variations are called, respectively, Lookahead and Lookahead-Monotone. In a
third variation, we allow the algorithm the knowledge not
only of the target’s interests, but also those of its neighbors’; this is a ‘reverse’ and limited form of lookahead, and
has precedent in Milgram’s experiment, where the sources
had the knowledge that the target was the wife of a student
of divinity in Cambridge, Mass. This is naturally aimed
at routing to hard-to-reach destinations by augmenting the
algorithm with extra information. The corresponding vari-
100
Success Rate
80
60
40
20
Lookahead Monotone
Local Monotone
Lookahead
Local
0
0
2
4
6
8
10
12
Minimum degree of the destinations
14
16
Figure 4: Success Rate with extended interests.
We briefly highlight some salient observations based on
Figures 3, 4, 5 and 6 and other related experiments.
(1) Navigation based on interests is an extremely powerful paradigm; the success of the basic algorithm Local
in achieving 21% successful routing is, a priori, unexpected,
given how crude our construction of the interest space is.
In particular the previous replicas of the small-world experiment had always lower successful rate [6, 30].
(2) Adding even one of two natural cues to local routing
(either expanding the interests of the target or adding a
step of lookahead) is enormously powerful — with each cue
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March 28–April 1, 2011, Hyderabad, India
tified the node of highest degree along successful paths, and
computed the average and median of its degree. While the
average degree of author nodes in the co-authorship network
is 6.7, the average and median values of the degree of the
node with the most connections along shortest paths are,
respectively, 133 and 163. For Lookahead-Expand, our
most successful decentralized routing algorithm, these values are, respectively, 189 and 228. These findings reinforce
the arguments of Granovetter [12] concerning the strength
of weak ties, as well as our analytical results proving the
importance of core nodes for decentralized routing.
Average path length without expanded interests
30
Lookahead Monotone
Local Monotone
Lookahead
Local
Path Length
25
20
15
10
5
0
0
2
4
6
8
10
12
Minimum degree of the destinations
14
16
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Figure 6: Average path length with extended interests.
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